441 research outputs found

    Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs

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    In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F\mathbf{F}, any infinite sequence M1,M2,...M_1,M_2,... of (skew) symmetric matrices over F\mathbf{F} of bounded F\mathbf{F}-rank-width has a pair i<ji< j, such that MiM_i is isomorphic to a principal submatrix of a principal pivot transform of MjM_j. We generalise this result to σ\sigma-symmetric matrices introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs, arXiv:0709.1433v4]. (Skew) symmetric matrices are special cases of σ\sigma-symmetric matrices. As a by-product, we obtain that for every infinite sequence G1,G2,...G_1,G_2,... of directed graphs of bounded rank-width there exist a pair i<ji<j such that GiG_i is a pivot-minor of GjG_j. Another consequence is that non-singular principal submatrices of a σ\sigma-symmetric matrix form a delta-matroid. We extend in this way the notion of representability of delta-matroids by Bouchet.Comment: 35 pages. Revised version with a section for directed graph

    Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices

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    We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_j, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour's theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum's theorem for graphs of bounded rank-width with respect to pivot-minors.Comment: 43 page

    When Can Matrix Query Languages Discern Matrices?

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    We investigate when two graphs, represented by their adjacency matrices, can be distinguished by means of sentences formed in MATLANG, a matrix query language which supports a number of elementary linear algebra operators. When undirected graphs are concerned, and hence the adjacency matrices are real and symmetric, precise characterisations are in place when two graphs (i.e., their adjacency matrices) can be distinguished. Turning to directed graphs, one has to deal with asymmetric adjacency matrices. This complicates matters. Indeed, it requires to understand the more general problem of when two arbitrary matrices can be distinguished in MATLANG. We provide characterisations of the distinguishing power of MATLANG on real and complex matrices, and on adjacency matrices of directed graphs in particular. The proof techniques are a combination of insights from the symmetric matrix case and results from linear algebra and linear control theory

    Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

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    In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time O~(mlogκlog2(1/ϵ))\widetilde{O}\left(m\log \kappa \log^2 (1/\epsilon)\right) where ϵ\epsilon is the amount of error we are willing to tolerate. Here, κ\kappa represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever κ\kappa is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time O~(m3/2log(1/ϵ))\widetilde{O}(m^{3/2} \log (1/\epsilon)). In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201

    Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model

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    Starting from the Verma module of U_q sl(2) we consider the evaluation module for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Sch\"{u}tzenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin and Kitanine. Here we establish that both generating functions satisfy Baxter's TQ-equation in the crystal limit by tying them to special U_q sl(2) solutions of the Yang-Baxter equation. The TQ-equation amounts to the well-known Jacobi-Trudy formula leading naturally to the definition of noncommutative Schur polynomials. The latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry: it is isomorphic to the fusion ring of the sl(n)_k -WZNW model whose structure constants are the dimensions of spaces of generalized theta-functions over the Riemann sphere with three punctures.Comment: 24 pages, 6 figures; v2: several typos fixe

    The su(n) WZNW fusion ring as integrable model: a new algorithm to compute fusion coefficients

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    This is a proceedings article reviewing a recent combinatorial construction of the su(n) WZNW fusion ring by C. Stroppel and the author. It contains one novel aspect: the explicit derivation of an algorithm for the computation of fusion coefficients different from the Kac-Walton formula. The discussion is presented from the point of view of a vertex model in statistical mechanics whose partition function generates the fusion coefficients. The statistical model can be shown to be integrable by linking its transfer matrix to a particular solution of the Yang-Baxter equation. This transfer matrix can be identified with the generating function of an (infinite) set of polynomials in a noncommutative alphabet: the generators of the local affine plactic algebra. The latter is a generalisation of the plactic algebra occurring in the context of the Robinson-Schensted correspondence. One can define analogues of Schur polynomials in this noncommutative alphabet which become identical to the fusion matrices when represented as endomorphisms over the state space of the integrable model. Crucial is the construction of an eigenbasis, the Bethe vectors, which are the idempotents of the fusion algebra.Comment: 33 pages, 8 figures; published in conference proceedings (Kokyuroku Bessatsu) of the workshop "Infinite Analysis 10, Developments in Quantum Integrable Systems", Research Institute for Mathematical Sciences, Kyoto, June 14-16, 2010; v2: some typos in Section 4 fixe

    Polygon gluing and commuting bosonic operators

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    We construct two series of commuting Hamiltonians parametrized by a constant matrix. The first series was my guess and the second one was know, and in our approach follows from the first series. For the proof we use familier facts known from our previous consideration of the links between random matrices and Hurwitz numbers, however the text is self-consistent.Comment: 9 pages, 1 figur
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